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Full text of "Mathematical And Physical Papers - Iii"

ticular, the isothermal curves on the two faces are ellipses of the kind just mentioned*.    Hence
(1)    If the plate be cut in a direction perpendicular to one of the thermic axes, the line joining the centres of a pair of ellipses which correspond on the two faces to a given temperature (such as that of melting wax) will be normal to the plate.    The principal axes of the ellipses will be in the direction of the two remaining thermic axes, and will be proportional to the square roots of the corresponding conductivities.
(2)    If the normal to the plate be not a thermic axis, the line joining the centres of the ellipses will be inclined to the normal, its direction being determined as above explained.
(3)    If the plate be cut in a direction parallel to either of the circular sections of the thermic ellipsoid (the three principal conductivities being supposed unequal,) the isothermal curves on both faces will be circles, but the line joining the centres of the two systems of circles will be inclined to the normal.
If the heat be communicated uniformly along the line of sources, or if there be only a single source situated midway between the faces, or more generally if the sources be alike two
* The problem solved in this article forms a good example of the advantage of considering the auxiliary solid. In M. Duhamel's memoir the plate is regarded as extremely thin, so that the variation of temperature in passing from one point to another of the same normal may be considered insensible—and it is remarked that the second case (in which a normal to the plate is not a thermic axis) is much more difficult than the first; whereas here the plate is not necessarily thin, and both cases follow immediately from what with regard to an uncrystallized body is self-evident. M. Duhamel has shewn that the isothermal curves on the two faces are ellipses, having thoir principal axes parallel and proportional to those of the ellipses in which the thermic ellipsoid is cut by planes parallel to the faces of the plate: but his demonstration that the line joining the centres of the two systems of ellipses has the direction assigned in the text does not seem altogether satisfactory, because the analysis only applies to the case in which the thickness of the plate is regarded as indefinitely small; whereas the space by which the ellipse corresponding on one face to a given temperature overlaps the ellipse corresponding on the other face to the same temperature is a small quantity of the order of the thickness of the plate, and ought for consistency's sake to be neglected.
The results contained in the remaining part of this paper are not found in the memoirs of M. Duhamel. It may here be remarked, that the results arrived at by the consideration of the auxiliary solid, such for example as that of Art. 17, might have been obtained by referring the crystal to oblique axes parallel to a system of conjugate diametei'K of the thermic ellipsoid.